[1] What type of clause is the bracketed part in the following sentence and what grammatical function does it assume? I'm sure [that you'll know a lot of people there].

[2] I also have problem figuring out the grammatical function of the that clause in the following sentence. "Dutch police are investigating the possibility [that a bomb was planted on the jet]". Is it a relative clause?

They are both declarative content clauses functioning as complements of "sure" and "possibility" respectively.

Your second example can't be a relative clause because there is no (overt or covert) relativized element anaphorically linked to the head noun "possibility". The subordinate clause simply expresses the content of the possibility.